Converting from Probabilities to Odds

If you know the probability of an event (p), the odds against the event occurring are [(1/p) -1]. For the example of the flush, 1 divided by 0.2 equals 5, subtract 1 to get 4, and the odds are 4 to 1. For unlikely events, the probabilities are small, which means 1/p is a large number and is very close to the odds against the event happening. For example, the probability of receiving two Aces for pocket cards is 0.004525 or 0.4525%. That means 1/p is 221, so the odds against receiving two Aces are 220 to 1. On average, for every hand with two Aces, there are 220 without. For events that are much more likely, 1/p is not very close to the odds. The probability of completing a flush with two cards to come, when you have a four-flush, is 0.35 or 35%. In this case, 1/p is 2.86, so the odds are 1.86 to 1 (almost 2 to 1). You will have about 2 failures for each success when drawing to flush if you already have four of the cards.

In the charts that follow, the frequencies of events are expressed as probabilities in some cases and odds against in other cases. The probabilities are expressed in percentages. For example, if you start with a pocket pair, 71.84% of the time your five-card hand after the flop will be one-pair. The remaining 38.16% of flops will improve your hand to   better than one pair (two pair, trips, full house or quads). For frequent events, percentages are a useful way of thinking. But for remote events, it is often easier to remember the odds. The odds against being dealt pocket Aces are 220-1, while for Ace-King, the odds against are 82-1.